I'm not seeing a current picture witht he sink -- just the one with the paper down on the wood which I understand to be before use. That wall looks like it has a raw end of drywall, not a side that is finished (paper or whatever) or maybe even no drywall. It looks like they tried to smooth joint compound in there, either by itself over your studs and built up or over something less than a full side of finished drywall. Joint compound will turn to mud (that's what they call it when it's wet -- hence the verb to "mud" the walls) and dissolve away if it gets wet. Paint will only seal the surface, and if water gets behind it -- well, I dare say it is probably worse than damp sheetrock, which is worse than damp green board. You say they had to sheer away your window sill. Did that sill come below the window too? I don't think you've got clean drywall there. I think you've got a really poor job of mudding and an even worse job of sealing the joint. That's my guess.

I'll go with Brickeyee's method...quicker, faster, easier. Most of all its the better one. Use a proper carpenter's knife or a lino knife to cut along the bottom and top then use a drywall saw to come down each stud. (buy a couple or three drywall saws, you will need them) You only have to smack out the tiles that are in the way of the up/down cuts. Less broken tile bits flying about, the better. (I'm assuming you don't have power tools...me, I'd tape/cover/protect the counter top, etc*. and use a recip saw with short blades...those blades will have Demo stamped or marked on them, wood blades won't work very long)((Actually the Demo blades may not either, depends on if its mastic or thinset right now)) I'll also back up his suggestion to lay ply or at least hardboard on the counter tops. *I'd also bag the upper cabs, paper the lowers and use a movable sheet of ply or board right under where I was actively smacking tile, lean this up against the lower cab faces, between you and the cabinets. I'd have the floor covered with paper and mover blankets in that order at the least. Then a sheet and a half of cement board re-installed where the drywall used to be, set your electrical boxes back in, thin set, tile, grout, caulk, done. Going over the old drywall is not a long term fix. Two day affair and that only to allow the thinset to set up overnight. If you spring for modified thinset with accelerated dry time you could do it in a long day. I know mastic works but I use thinset anywhere water is present. Even a backsplash.

cigi, in our last house the previous owner installed tumbled marble in a mauvish tone which IMO looked awful with the St Cecilia granite. Even though the tile was expensive, I tore them off and left the 4" granite backsplash, patched and painted the rest of the wall. The sink was the only working area as the cooktop was on the island. No problem with water as I used BM Aura, but Pittsburgh Paint's Manor Hall Timeless is just as good with a more flat look and lifetime warranty. I'm pondering the issue for my current redo. In various houses I've had paint, wallpaper, white tile with abalone inserts, plain white tile. Right now I'm weighing white tile with a beautiful insert, just paint with no granite backsplash since this area has only a counter but no cooking/wet functions, paint with granite backsplash, or a simple glass tile backsplash with listello over it a la Chinchette. I'm attaching a photo from Chinchette's beautiful kitchen (hope she doesn't mind seeing hers cited as a great example) which shows the last option. In any event, I echo others' advice to not worry about that decision immediately unless you absolutely love a certain look. Just use good paint. Here is a link that might be useful:

So tired of hearing " but that will make it look dated" or "it wouldn't be good for resale" Everything is going to be dated at some point, the kitchen designers, cabinet companies and appliance companies make sure of that. Example: stainless steel appliances were just coming on the market 15 years ago when we gutted our kitchen for a remodel. It was hard for me to find stainless in my area at that time, I live in a small mid western town. So, have you noticed in the past few years appliance companies have been trying to bring out new colors, grayed looking stainless, now ice white etc. they are trying to make people think their stainless appliances are soon going to be "dated". But I have read in several articles that stainless is still the number one choice. I chose a cinnamon stained cherry wood for our new build. I thought about white because that is the trend right now, but I reasoned cherry cabinets will always be in style while the white trend Has just a few more years. I won't be a slave to trends. Just my opinion.